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	\textbf{{\Huge self-reflectiveness and L-systems for polarized-fractional light algebras}}
{\center{Feb 4, 2011, 
	Johan Ceuppens, Theo D'Hondt - Vrije Universiteit Brussel}}
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{\large{\textbf{\center Abstract}}}
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\section{\large Introduction : Group Theory and Doppler shift}

A Lie group gets a differentiable manifold, which means the limit exist for a derivative instead of just the limit 
(because of compact Minkwoski space)
The Lorentz group is a linear space/transformation Lie group with homeomorphisms in the linear space itself  (e.g. group O(1,3) )
If one would extend the Minkowski space-time to higher dimensions the othrogonality principle holds but not the 3-dimensional
 orthonormality. Though spacetime can differentiate itself towards the Lorentz transformation (due to Doppler shifting of matter and field radiation.)
 As noted above, only the differential should exist not the linearity (orthnormality). The differential structure is due to 
homeomorphisms in linear space. 

Differentiability means different things in different            
contexts including: continuously differentiable, k times         
differentiable, and holomorphic. Furthermore, the ability        
to induce such a differential structure on an abstract           
space allows one to extend the definition of       
differentiability to spaces without global coordinate            
systems. -- wikipedia - diff. manifolds 
Tensor and vector fields apply a calculus to the Doppler shift with tubular manifolds for elementary particle interactions. 
Gauge bosons, e.g. photons travel with dark matter Doppler shifts. 
The Lorentz path integral for Hopfield NN thinking (self-reflective NN using light polarization as a neuron-synaps theory )
means that light exerts thinking by the photo-electric effect of neuron activations AND polarization of that light.)
Thus we have electro-magnetic thinking and dark matter gravitational Doppler shifts, dark matter thinking also due to
polarization and lensing effects. The lensing effects can be calculated with GAs on the lens dynamical system.
\section{\large Introduction : Group theory and Gauge Bosons }
A definition of bosons:                                                          
    1. photons,                                                     
    2. W and Z bosons, and                                          
    3. gluons.                                                  
                                                                    
   Each corresponds to one of the three Standard Model              
   interactions: photons are gauge bosons of the             

The Lorentz transfo attracts as it is a root, thus spacetime can be curved (spinors and tensors.)
Structure of the Higgs field :  
                                                                    
   In the standard model, the Higgs field is an SU(2) doublet,      
   a complex spinor with four real components (or equivalently      
   with two complex components), with a Standard Model U(1)         
   charge of -1. )   It transforms as a spinor under SU(2).
	(nvdr. Higgs field is a virtual particle , boson interaction mechanism)
Background :                                                  
                                                                    
   Spontaneous symmetry breaking offered a framework to             
   introduce bosons into relativistic quantum field theories.       
   However, according to Goldstone's theorem, these bosons          
   should be massless. The only observed particles which could      
   be approximately interpreted as Goldstone bosons were the        
   pions, whic

TODO: Higgs mechanism self-reflection.
Spinors -> complex vector space. object for determinig spin
(rotation groups-> 360 degrees, spin changes, 720 degrees, no change)

The Lorentz transfo is a path integral, energy function and NN classificator. It is a root with refractive properties e.g. light.
Electronics use Lorentz transfos using logical adder circuits.

The spinor can be seen as a selfreflection in Hopfield networks but this is not everything needed to think. Other  systems to think may exist as shown with field quantum mechanics, tensor fields and so on.

The Lorentz transfo is a quantum computational system by group theory. Quantum Lorentz encryption can be used to develop a 
cryptofier or code breaker by jumping between quantum states and using GA properties of the Lorentz group.
TODO: quantum encryption uses photon polarization

We could try to develop a Quantum Group Theory, non-repetitiveness and self-reflection in nature.
Fibonnaci groups for example can be used as a row of functions using Banach spaces.

Hopfield NN: 1. self-reflective (spinor 1/2 180 and 360 degrees)
		  2. function based (sigmoid activation function, neuron firing through synaps)
		  3. self-reflection with function (activation)
Try to build a chaos theoretical machine / dynamical system out of the above 3 points.

L-systems + self-reflectiveness in nature
Higher order quantum computation languages can use these Hopfield classificators (i.e. the group theory) 
TODO: K theory
Perturbation theory, self-reflective neuron in Hopfield NN as an NN bias, solutions for the Shrodinger equation 
are not always there the Hamiltonian is too complex (the calculation of the Hamiltonian is only possible with a good 
Hamiltonian operator laplacian, see defnition of Shrodinger equation, which is a time-space (in)dependent equation.)

The laplacian calculation of the Hamiltonian operator (Epsi=Hpsi)
	of the Shrodinger equation 
	Energy gets calculated by solving the Hamiltonian
	operator on a wave function psi
	(solutions to 2nd order partial diff eqs.)


The Shrodinger equation has been shown through experiment it cannot 
be derived theoretically.
It describes how the wave function of a physical
system evolves over time
(Hamilt. matrix properties : 
The transpose of a Hamiltonian matrix is Hamiltonian.
The trace of a Hamiltonian matrix is zero.
The commutator of two Hamiltonian matrices is Hamiltonian.
!!! The eigenvalues of any Hamiltonian matrix are symmetric about the imaginary axis !!!.
The space of all Hamiltonian matrices is a Lie algebra. 

A linear operator is Hamiltonian with respect to ? if and only if its matrix in this basis is Hamiltonian

Since electrons are fermions, the Pauli exclusion principle forbids them from occupying the same quantum state, so electrons 
have to "stack" within an atom, i.e. have different momenta while at the same place.

\section{\large Conclusion}


\bibliographystyle{plain}
\bibliography{refs} % refs2.bib

%Ph. D. P. van Remortel - VUB 

%Cybernetics 2nd ed. - N. Wiener

%M. thesis - Johan Ceuppens

%Orgins of Order - book Kauffman

%Graphics Gems 123 - book

%Neural Computers - book

%article Zhou
%article Savvides 
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